Synthetic controls methods to estimate the causal effect(s) of an intervention across multiple treated units

Carlos G. Poses

The effect of a policy intervention

The effect of a policy intervention

The effect of a policy intervention

We need counterfactuals

A possible causal effect

A possible causal effect

A possible causal effect

A possible causal effect

Another possible causal effect

Another possible causal effect

Another possible causal effect

Estimating the counterfactual

Using auxiliary data to reconstruct the counterfactual

Using auxiliary data to reconstruct the counterfactual

Using auxiliary data to reconstruct the counterfactual

Synthetic control = estimated counterfactual

Synthetic control = estimated counterfactual

Synthetic control = estimated counterfactual

Synthetic control = estimated counterfactual

Synthetic control = estimated counterfactual

Behind the scenes: weights estimation

  • Synthetic control: weighted average of other units

  • Estimation…

    • is complex
    • is sparse
    • ‘Matching’ on predictors

Synthetic control



Simply a weighted average!


\begin{align*} &\text{Synthetic California}_{t} = \\ &\quad w_{\text{Nevada}} \cdot \small{\text{Sales}}_{\text{Nevada}} + \\ &\quad w_{\text{Connecticut}} \cdot \small{\text{Sales}}_{\text{Connecticut}} + \\ &\quad w_{\text{Colorado}} \cdot \small{\text{Sales}}_{\text{Colorado}} + \\ &\quad w_{\text{Montana}} \cdot \small{\text{Sales}}_{\text{Montana}} + \\ &\quad w_{\text{Utah}} \cdot \small{\text{Sales}}_{\text{Utah}} \end{align*}

‘Matching’ on predictors

California
0.17
24.28
10.12


Predictor Colorado Connecticut Montana Nevada Utah
age15to24 0.16 0.16 0.15 0.15 0.18
beer 25.08 20.70 27.88 37.00 13.34
lnincome 10.00 10.27 9.76 10.05 9.71

‘Matching’ on predictors

California
0.17
24.28
10.12


=

Predictor Colorado Connecticut Montana Nevada Utah
age15to24 W_Col*0.16 W_Con*0.16 W_Mon*0.15 W_Nev*0.15 W_Utah*0.18
beer W_Col*25.08 W_Con*20.7 W_Mon*27.88 W_Nev*37 W_Utah*13.34
lnincome W_Col*10 W_Con*10.27 W_Mon*9.76 W_Nev*10.05 W_Utah*9.71
  • Where w_j > 0 and \sum{w_j = 1}

Where to go to know more?

The synthetic control was developed by Alberto Abadie and collaborators.

  • The best resource to understand synthetic controls i the overview paper by Abadie (2021).

  • Other nice resources are Abadie, Diamond, and Hainmueller (2010) (including the proof that the estimator is unbiased) and Abadie and Gardeazabal (2003) (the first paper about the method).

How to fit a synthetic control model?

It’s pretty easy with the tidysynth package.

library(tidysynth)

freq_model <- synthetic_control(data = prop99, 
                                outcome = cigsale,  
                                unit = state, 
                                time = year,  
                                i_unit = "California",  # treated state
                                i_time = 1988) # time of the intervention

Note

  • If you want to take a peak at how the data look like, you can check it here:

Two goals of the thesis

Goal 1: A valid estimate of uncertainty

Goal 2: A method that works with multiple treated units

Goal 2: A method that works with multiple treated units

Work so far

  • Bayesian synthetic control for goals 1 & 2

  • So far, goal 1:

    • In-depth literature review
    • Single treated unit Stan programming

Preliminary simulation results


  • Data generated with a multiple regression linear model unobserved and observed (normal) predictors

  • Additive causal effect (+400 at all time points)

Simulation results
Method Standard Deviation Bias MSE Coverage
Maximum a Posteriori 47 0.98 2,252 -
Posterior mean 92 3.73 8,472 0.24
Current implementation 45 1.81 2,001 -

Next steps

  • Modelling:

    • Introduce a penalty
      • Lower variance
      • Similar MAP and posterior mean
    • Sparsity priors

  • More simulation conditions, including time-varying coefficients

  • Goal 2: Multiple treated units

End



  • Thanks to my supervisors Oisín Ryan & Erik-Jan van Kesteren

  • Thanks for listening!

References

Abadie, Alberto. 2021. “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature 59 (2): 391–425. https://doi.org/10.1257/jel.20191450.
Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2010. “Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program.” Journal of the American Statistical Association, June. https://doi.org/10.1198/jasa.2009.ap08746.
Abadie, Alberto, and Javier Gardeazabal. 2003. “The Economic Costs of Conflict: A Case Study of the Basque Country.” American Economic Review 93 (1): 113–32. https://doi.org/10.1257/000282803321455188.